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Research Papers: Gas Turbines: Aircraft Engine

Inversion of the Fundamental Thermodynamic Equations for Isentropic Nozzle Flow Analysis

[+] Author and Article Information
Joseph Majdalani

H. H. Arnold Chair of Excellence in Advanced Propulsion Professor maji@utsi.edu

Brian A. Maicke

Mechanical, Aerospace and Biomedical Engineering Department,  University of Tennessee Space Institute, Tullahoma, TN 37388

J. Eng. Gas Turbines Power 134(3), 031201 (Dec 28, 2011) (9 pages) doi:10.1115/1.4003963 History: Received February 22, 2011; Revised April 02, 2011; Published December 28, 2011; Online December 28, 2011

The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area expansion and specific heat ratios. These fundamental thermofluid relations are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the possible solutions are systematically identified and exacted. Furthermore, for each branch of solutions, two types of recursive approximations are provided: a property-specific formulation and a more general, universal representation that encompasses all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from Bürmann’s theorem of classical analysis. Bosley’s technique is then applied to verify the theoretical truncation order in each approximation. The final expressions enable us to estimate the pressure, temperature, and density for arbitrary area expansion and specific heat ratios with no intermediate Mach number calculation or iteration. The analytical framework is described in sufficient detail to facilitate its portability to other nonlinear and highly transcendental relations where closed-form solutions may be desirable.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Variable area duct showing relevant thermodynamic properties and physical stations

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Figure 2

Numeric and asymptotic solutions for the subsonic (a) pressure and (b) temperature

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Figure 3

Performance comparison between (a) pressure specific and universal approximations for the first two values of n and (b) the percent relative error in the subsonic solution at increasing asymptotic orders

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Figure 4

Numeric and asymptotic solutions for the supersonic (a) pressure and (b) temperature

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Figure 5

Percent relative error in the supersonic pressure, temperature, and density ratios based on the universal asymptotic representation

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Figure 6

Absolute error in the supersonic solution using the universal representation

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